_7_ "_r 3007

- - August 1990

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= r 2 _ '

Buckling and Postbuckling

Behavior of Square

Compression'Loaded

Graphite-Epoxy Plates

With Circular Cutouts

Michael P. Nemeth
m

. ._ _.2 -

2- _.

|
NASA

Technical

Paper

3007

1990

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Division

Buckling and Postbuckling

Behavior of Square

Compression-Loaded

Graphite-Epoxy Plates

With Circular Cutouts

Michael P. Nemeth

Langley Research Center

Hampton, Virginia

Summary

An experimental study of the postbuckling behavior of square compression-loaded graphite-epoxy

plates and isotropic plates with a central circular

cutout is presented. Results are presented for unidirectional [010]s and [9010]s plates, [(0/90)5]s plates,

and for aluminum plates. Results are also presented

for [(+0)6]s angle-ply plates for values of 0 = 30 °,

45 ° , and 60 ° .

The experimental results indicate that the change

in axial stiffness of a plate at buckling is strongly

dependent upon cutout size and plate orthotropy.

The presence of a cutout gives rise to an internal

load distribution that changes, sometimes dramatically, as a function of cutout size coupled with the

plate orthotropy. In the buckled state, the role of orthotropy becomes more significant, since bending in

addition to membrane orthotropy is present. Most

of the plates with cutouts exhibited less postbuckling stiffness than the corresponding plate without a

cutout, and tile postbuckling stiffness decreased with

increasing cutout size. However, some of the highly

orthotropic plates with cutouts exhibited more postbuckling stiffness than the corresponding plate without a cutout. These results suggest that the complex

interaction of cutout size and plate orthotropy on

the internal load distribution in plates needs further

investigation. These results also suggest the possibility of tailoring the cutout size and the stacking sequence of a composite plate to optimize postbuckling

stiffness. An important finding of this experimental

study is that plates with large radius cutouts exhibit

some postbuckling strength.

The experimental results also indicate that a

cutout can influence modal interaction in a plate.

Specifically, results are presented that show a plate

with a relatively small cutout buckling at a higher

load than the corresponding plate without a cutout

due to modal interaction. Other results are presented

that indicate the presence of nonlinear prebuckling

deformations, due to material nonlinearity, in the

angle-ply plates with 0 = 45 ° and 60 °. The nonlinear prebuekling deformations are more pronounced

in the plates with 0 = 45 °, and become even more

pronounced as the cutout size increases. Results are

also presented that show how" load-path eccentricity

due to improper machining of the test specimens affects the buckling behavior. Some of the plates with

cutouts and eccentricity exhibited a snap-through

type of buckling behavior.

Introduction

The technical challenges associated with the development of military and civilian aerospace vehicles

for the 21st century have identified several key areas that need further development. One important

area is the design technology and analysis of largescale composite structures. The high performance

requirements of these structures have led to a search

for ways to exploit their tailorability to meet specific

mission goals.

An important structural component used in practically all aerospace vehicles is the rectangular plate

with a central circular cutout. Cutouts commonly

appear in plates as access ports for mechanical and

electrical systems or are included to reduce the structural weight in components such as wing ribs and

spars. Often during flight, these members experience

compression loads, and thus their buckling and postbuckling behavior are important factors that nmst be

considered in their design.

Investigations of the buckling behavior of plates

with cutouts have appeared in the technical literature since 1943. A summary of these investigations,

for both isotropie and lmninated composite plates,

is given in reference 1. In-depth parametric studies

of the buckling behavior of square and rectangular

plates with central circular cutouts are presented in

references 1 through 3. Analytical and experimental

results are presented in these studies that indicate

buckling behavior trends for a wide range of plate parameters. The results and physical insight presented

in these references indicate that the buckling behavior of compression-loaded isotropic and orthotropie

plates with cutouts is well understood.

Substantially fewer studies on tile postbuckling

behavior of plates with cutouts are available in the

technical literature. Some of the first studies were

presented by Yu and Davis in reference 4, by Martin

in reference 5, by Yu and Davis in reference 6, and by

Ritchie and Rhodes in reference 7. The results presented in references 4 and 6 address the postbuekling

collapse of steel beams, columns, and plate girder

structures with cutouts in their webs. The results

presented in references 5 and 7 focus specifically on

square isotropie plates with central circular cutouts.

In addition, buckling and postbuekling results are

also presented in reference 5 for square laminated

composite plates with central circular cutouts.

More recently, selected results for the postbuckling and failure characteristics of compression-loaded

rectangular graphite-epoxy plates with central circular cutouts have been presented in reference 8. Additional recent studies of the postbuckling collapse

of square isotropic plates with square and circular

cutouts are presented in references 9 and 10. A

study of the imperfection sensitivity and postbuckling strength of compression-loaded square isotropic
andlaminatedcompositeplateswith centralcircular

cutoutsispresentedin reference11.

Reviewof the studiespresentedin references4

through11indicatesthat theeffectsof cutoutsize,

plateaspectratio, and laminatestackingsequence

on the postbucklingbehaviorof platesarestill not

wellunderstood.Thispaperexaminesthe behavior

ofselectedisotropicandgraphite-epoxycompressionloadedsquareplatesandattemptsto establishoverall

trendsindicatingtheeffectsof cutoutsizeandplate

orthotropyonplatepostbucklingbehavior.Thepaper focuseson an experimentalstudy of unidirectional[0m],_and[90m]._plates,[(0/90)5]splates,and

alunfinumplates. Theseplatesrepresentextreme

easesof orthotropyandmoderateorthotropyaswell

as isotropy.Resultsarealsopresentedfor [(+0)6]._

angle-plyplatesforvaluesof 0 = 30 °, 45 °, and 60 °.

Symbols

D11, D22

d

H

orthotropic plate bending stiffness

hole diameter, in. (see fig. 2)

nominal plate thickness, in. (see

table 8)

K buckling coefficient, PcrW

7r 2 D_77D22

L plate length, in. (see fig. 2)

P axial load, lb

Pc; ,axial load at buckling, lb

P_?r axial load at buckling for d/W = 0

case, lb (see table 8)

W plate width, in. (see fig. 2)

A end-shortening, in. (see fig. 2)

Acr end-shortening at. buckling for

d/W = 0 ease, in. (see table 8)

0 fiber orientation angle (see fig. 4)

5 transverse deflection at. edge of

cutout, in.

Specimens, Apparatus, and Tests

The aluminum specimens tested in this investigation were machined out of 6061-T6 aluminum sheets

having a nominal thickness of 0.0625 in. Several

thickness measurements were made on each specimen, and the average thickness was determined to

be 0.0647 in. Nominal material properties were assumed to include a Young's modulus E of 11.0 x 106

psi and a Poisson's ratio u of 0.33.

The composite specimens tested in this investigation were fabricated from commercially available

450 K (350°F) cure Hercules AS4/3502 graphiteepoxy preimpregnated tapes. Nominal lamina properties were assumed to include a longitudinal modulus E1 of 18.5 x 106 psi, a transverse modulus E2

of 1.6 x 106 psi, an inplane shear modulus G12 of

0.832 x 106 psi, a major Poisson's ratio u12 of 0.35,

and a nominal ply thickness of 0.005 in. The tapes

were laid up to form 20-ply-thick laminates having [010Is, [(0/90)5]s, and [90101s stacking sequences

and to form 24-ply-thick laminates having [(+30)6]s,

[(-4-45)6]_, and [(+60)6]_ stacking sequences.

The laminates were cured in an autoclave with

the manufacturer's recommended procedures. After

curing, the laminates were ultrasonically C-scanned

to assess specimen quality and then machined into

test specimens. All specimens were 10 in. long by

10 in. wide, and the loaded edges were machined

fiat and parallel to permit uniform compressive loading. Central circular cutouts were machined into the

aluminum panels with a milling machine and machined into the composite panels with diamond impregnated core drills. The circular cutout diameters

ranged from 0 to 6.25 in. One side of each specimen

was painted white to reflect light so that a moirdfringe technique could be used to monitor the outof-plane deformations. A total of 40 specimens were

tested. The specimens have the following designations: A1 through A7 for the aluminum specimens,

B1 through B7 for the [Ol0Js specimens, C1 through

C6 for the [9010Is specimens, D1 through D7 for the

[(0/90)5]s specimens, El through E5 for the [(+30)6]s

specimens, F1 through F4 for the [(±45)6]s specimens, and G1 through G4 for the [(+60)6]s specimens. The cutout sizes for each specimen are given

in tables 1 through 7. Several thickness measurements were also made on each composite specimen.

The average thickness values were determined to be

0.107 in. for the [0m]s laminates, 0_106 in. for the

[90m]s laminates, 0.110 it,. for the [(0/90)5]s laminates, 0.1176 in. for the [(±30)6]s and [(±60)6]s laminates, and 0.1307 in. for the [(±45)6]_ laminates.

The specimens were loaded gradually in axial

compression with a 300-kip-capacity hydraulic testing machine. The loaded ends of the specimens were

clamped by fixtures during testing, and the unloaded

edges were simply supported by restraints that prevent the specimen from buckling as a wide cohmm.

Most specimens were loaded to approximately twice

the buckling load, and then the test was stopped.

Some specimens were loaded until failure. A typical specimen mounted in the test fixture is shown in

figure 1.

Electrical resistance strain gauges were used to

measure strains, and direct-current differential transformers were used to measure axial displacements
anddisplacementsnormalto thespecimensurface.

Electricalsignalsfromtheinstrumentationandtile

correspondingappliedloadswererecordedon magnetictapeat regulartimeintervalsduringthetests.

Results and Discussion

Resultsarepresentedfor severaldifferentaluminumandgraphite-epoxyplates.First,the methodsusedto obtain the experimentalvaluesof the

prebucklingstiffnesses,bucklingloads,and initial

postbucklingstiffnessesarepresented.Resultsare

thenpresentedfor the aluminumplates;tile [010Is,

I9010]s,and [(0/90)5]sspeciallyorthotropicplates;

andthe [(+30)61s,[(±45)61s, and[(4-60)6]ssymmetrically laminatedangle-plyplatesin separatesubsections.Failureresultsarethenpresentedfor the

specimensthat wereloadedto failure. After these

subsections,resultsindicatingtile overallstiffness

trendsexhibitedby thespecimensandresultsindicatingtheeffectsofcutoutsizeandplateorthotropy

onthedistributionof theout-of-planedisplacement

fieldsin thespecimensarepresented.Finally,results

showingmodalinteractionin a platewith a cutout,

andresultsindicatingtheeffectsof load-patheccentricity duc to impropermachiningof someof the

specimensarepresented.

Analysis and Representationof TestData

To illustratetile postbucklingbehaviorof plates

with cutouts,curvesof nondimcnsionalload versusend-shorteningandnondimensional loadversus

transversedeflectionarepresentedinthispaper.The

curveof nondimensional loadversusend-shortening

for eachspecimenwasobtainedby first performing

a least-squaresfit of a straightline to tile mostlinearpart of theprimarybranchof theplot of actual

loadversusend-shorteningrecordedduringthetest.

With theequationofthelineobtainedfromtheleastsquaresfit of thetestdata,theprebueklingstiffness

wasobtaineddirectly,andthe initial irregularityin

theplotof actualloadversusend-shortening, associatedwith initial slackin thetestfixture,waseliminated.Theeliminationwasperformedbytranslating

thecoordinatesystemof theplotof actualloadversusend-shorteningsuchthat thelineobtainedfrom

theleast-squaresfit passedthroughtheorigin.

In asimilarmanner,theinitial postbucklingstiffnesswasobtaineddirectly by performinga leastsquaresfit of a straightlineto themostlinearpart

of the secondarybranchof the plot of loadversus

end-shorteningrecordedduringtesting.Theexperimentalbucklingloadandassociatedend-shortening

werethenobtainedby computingtheintersectionof

thetwostraightlinesfitted to the primaryandsecondarybranchesof thetestdataforloadversusendshortening.

Finally,the curveswerenondimensionalizedby

dividingtheloadandend-shorteningofagivenspecimenby analyticalvaluesof the bucklingload po

and end-shortening Ac°r, respectively, of the corresponding plate without a cutout. These analytical values were obtained with an in-house computer program for buckling analysis and are based

on the nominal material properties given in this

paper previously and on nominal plate thicknesses.

The nominal plate thicknesses used in the buckling calculations were based on tile average values

of the measured plate thicknesses previously described. Moreover, the analytical values were based

on square-plate geometry, and on a uniaxial loading condition in which two opposite edges of the

plate are uniformly displaced toward one another.

The loaded edges were assumed to be clamped, and

the mfloaded edges were assumed to be simply supported. In all the calculations, the buckling load

and corresponding end-shortening were based on the

9.5-in. unsupported length between knife edge supports of the test fixture. The nominal thicknesses

used in these calculations and tile corresponding analytical values are given in table 8.

The results for experimental load versus endshortening are presented in subsequent sections of

the paper by two straight lines obtained from the

least-squares fits to the primary and the initial part

of the secondary, branches of the test data. The

intersection of the two straight lines is marked with

a solid circular symbol ill tlie figures to denote the

experimental buckling load. The nondimensionaI

transverse deflection shown in the figures presented

herein was obtained by dividing the actual transverse

deflection .by the nominal plate thickness that was

used to compute po and Ac°r.

Differences between the analytical values and experimental values of the buckling loads and displacements of the plates without cutouts are noted. These

differences are attributed to differences between the

actual thicknesses of the plates and assumed nominal thickness used in the calculated values, and the

assumption of ideal clamped and simply supported

boundary conditions in the analysis. The differences

between analysis and experiment are also attributed

to the difference between the 9.5-in. unsupported

length between the ends of the test fixture that was

used as tile plate width in the buckling calculations

and the true plate width of 10 in.

The experimental results for some of the angle-ply

laminates exhibited secondary branches of the plots

of load versus end-shortening recorded during testing
that weretotally nonlinear.Thisattributemadeit

difficultto establishthe experimentalbucklingload

in themannerpreviouslydescribed.Forthesecases,

the experimentalbucklingloadwasestimatedfrom

the data for loadversustransversedeflectionand

from strain-gaugedata. The initial postbuckling

stiffnesswastakentobetheslopeofalinetangento

thesecondarybranchof theplotof loadversusendshorteningand intersectingthe straight-lineleastsquaresfit of theprimarybranchat theestimateof

thebucklingload.It is importantto point outthat

for thesecases,thevalueof theinitial postbuckling

stiffnessis sensitiveto the estimateof the buckling

load.

Aluminum Plates

Experimentalresultswereobtainedfor a square

aluminumplate without a cutout, and for square

aluminumplateshavingsix differentcutout sizes.

Thecutoutsize,bucklingload,prebucklingstiffness,

andpostbucklingstiffnessofeachplatearepresented

in table 1. Curvesfor nondimensional load versus end-shortening and nondimensional load versus

transverse deflection are presented in figures 2 and 3,

respectively.

The results presented in table 1 and figure 2 indicate that the prebuckling stiffnesses of the isotropic

plates decrease monotonically with increasing cutout

size. This trend is consistent with the fact that

an increase in cutout size gives rise to a decrease

in the cross-sectional area at the net section of the

plate. The maximum decre_e in prebuckling stiffness compared with the stiffness of the plate without

a cutout is approximately 42 percent for the plate

with d/W = 0.6.

The results in table 1 also indicate that tile buckling loads of the plates decrease at first and then tend

to increase with increasing cutout size. Analytical results indicating a similar trcnd for the buckling loads

are presented ill reference 1 and suggest that increase

in experimental buckling load with increasing cutout

size presented herein is not due entirely to scatter

in the test data. The buckling mode shapes for all

the plates consisted of one half-wavc along both their

length and width.

Additional results presented in table 1 and figure 2 indicate that the initial postbuekling stiffnesses

of the plates decrease monotonically wittl increasing

cutout size. The largest decrease is approximately

43 percent for d/VV = 0.6. Comparing the prebuckling stiffness with the postbuckting stiffness given in

table I for each specimen indicates that, as the cutout

size increases, the change in axial stiffness duc to

buckling varies between 35 and 40 percent for the

full range of cutout sizes.

The curves for nondimensional load versus transverse deflection shown in figure 3 give an indication

of the relative size of the initial imperfection in tile

geometry of each plate. Moreover, these results give

an indication of the nature of the postbuckling deformations near the cutout. For tile plates with d/IV

less than 0.6, the transverse deflection was measured

on the top edge of the cutout as indicated in figure 3. For the plate with d/IV = 0.6, the measurement was made on the right edge of the cutout.

The curves shown in this figure cannot be compared

directly since the location of the transverse deflection mee, surement is different for each value of d/W.

Itowever, the curves do give some indication of the

postbuckling deformation near the cutout, since tile

transverse deflection me_urements for two adjacent

cutout sizes (e.g., d/W = 0.3 and 0.4) are near one

another and the plates possess similar defornlation

shapes (one half-wave along their length and width).

Comparing the relative sizes of the transverse deflections of the specimens with d/tV = 0.5 and 0.6 suggests that as the cutout size increases, the amount

of bending in the top central region of the plate is

less than the amount of bending in the right central

region of the plate.

Specially Orthotropic Plates

Experimental results were obtained for unidirectional [010]s, unidirectional [9010]s, and cross-ply

[(0/90)5]s square plates having up to seven different cutout sizes ranging from d/W = 0 to 0.66.

The cutout size, buckling load, prebuckling stiffness,

and postbuckling stiffness of each [010]_, [9010]_, and

[(0/90)5]., plate are presented in tables 2, 3, and

4, respectively. Curves of nondimensional load versus end-shortening and nondimensional load versus

transverse deflection are presented in figures 4 and

5, respectively, for the [0tO]s plates; in figures 6 and

7, respectively, for the [9010]s plates; and in figures 8

and 9, respectively, for the [(0/90)5]s plates.

Results for [010]s plates. The rcsults presented

in table 2 and figurc 4 indicate that the prebuckling

stiffncsses of the [010]_ platcs decre_sc monotonically

with increasing cutout size, with the exception of

the d/W = 0.32 case. For this particular case, it

was found that the loaded edges of the specimen had

not been properly machined. This observation suggests that impropcr machining of the loaded edges of

the test specimen may have produced an edge effect

that increased the prebuckling stiffness (sec subsequent section entitled "Anomalous behavior"). The

maximum decrease in prebuckling stiffness is approximately 74 percent for the plate with d/W = 0.66.

4
Theresultspresentedintable2alsoindicatethat

thebucklingloadsof tile platesdecreasemonotonicallywith increasingcutoutsize.Analyticalresults

showingessentiallytim sametrendarepresentedin

reference2. The maximumreductionis approximately33percentfor the platewith d/W = 0.66.

The buckling mode shapes for all the plates consisted

of one half-wave along both tile plate lengths and

widths.

The results presented in table 2 and figure 4 also

indicate that tim initial postbuckling stiffnesses of the

[010]s plates decrease monotonically with increasing

cutout size, with the exception of the d/W = 0.11

and 0.32 cases. The largest decrease is approximately 24 percent for d/W = 0.66. Comparing the

prebuckling stiffness with the postbuckling stiffness

given in table 2 for each specimen indicates that as

ttle cutout size increases, the change in axial stiffness due to buckling decreases monotonically from

approximately a 76-percent stiffness reduction for tile

plate with d/W = 0 to a 31-percent stiffness reduction for the plate with d/W = 0.66.

The curves of nondimensional load versus transverse deflection shown in figure 5 indicate that the

plates with d/W = 0.21 and 0.11 had tile largest

initial imperfections. In addition, tile results in figure 5 suggest that, for tile buckled plates, tile region

adjacent to the cutout (where the transverse deflection was measured) generally becomes more flexible

in bending as the cutout size increases.

Results for [9Orals plates. The results presented in table 3 and figure 6 indicate that the prebuckling stiffnesses of the [9010]s plates decrease with

increasing cutout size, for the most part, with the

exception of the d/W = 0.42 case. For this particular case, the prebuckling stiffness was determined

to be approximately 16 percent higher than the prebuckling stiffness of the corresponding plate without

a cutout. Tile reduction in prebuckling stiffnesses

exhibited by the plates with d/W < 0.42 was less

than 10 percent of the prebuckling stiffness of tile

corresponding plate without a cutout. The maximum decrease in prebuckling stiffness is approximately 54 percent for tile plates with d/W = 0.66.

These results suggest that a complex interaction between the plate geometry and the degree of plate orthotropy may be present.

The results presented in table 3 also indicate

that the buckling loads of the [9010]s plates do not

decrease monotonically with increasing cutout size.

The buckling load of the plate with d/W = 0.11 is

approximately 9 percent higher than the buckling

load of the corresponding plate without a cutout.

The lowest buckling load, approximately 30 percent

of tile buckling load for d/W = 0, is exhibited by the

plate with d/W = 0.42. The plate with d/W = 0.66

buckles at a load approximately 18 percent less than

the buckling load for tile corresponding plate without

a cutout. Analytical results showing tile same trend

are also presented in reference 2 and suggest that

the unusual trend exhibited by tile plates is not due

entirely to scatter in tile test data.

hi addition, the results presented in table 3 and

figure 6 indicate that tile initial postbuckling stiffnesses of tile plates do not typically decrease with

increasing cutout size. In fact, the experimental results indicate initial postbuckling stiffnesses approximately 36 percent larger than the stiffness of the

corresponding plate without a cutout for the plate

with d/W = 0.21, and ms high as 56 percent larger

for the plate with d/W = 0.42. The plate with

d/W = 0.66 exhibited a stiffness reduction of approximately 43 percent of the initial postbuckling stiffness

of tile corresponding plate witilout a cutout. Comparing the prebuckling stiffness with the postbuckling stiffness given in table 3 for each specimen indicates that tile change in axial stiffness due to buckling

is typically less than that of the d/W = 0 case and

ranges between 28 and 54 percent of their respective

prcbuckling stiffncsses.

The curves of nondimensional load versus transverse deflection shown in figure 7 for the [9010is plates

indicate that the plates with d/W = 0.42 and 0.32

exhibit the largest initial imperfections. The results

in this figure also suggest that the region adjacent to

the cutout in the buckled plates with d/W > 0.11

generally becomes more flexible in bending as the

cutout size increases. The largest amount of flexibility is exhibited by the plate with d/W = 0.42.

The results presented in this figure for the plate

with d/B T = 0 indicate that the plate has a relatively large geometric imperfection and deforms out

of plane somewhat before moving in the opposite

direction to form a nodal line at the center of the

plate (where the transverse deflection was measured).

The results presented in this figure for the plate with

d/W = 0.11 indicate that the plate has a very small

geometric imperfection and that the central region of

the plate deforms initially out of plane as if to form a

distinct mode shape and then moves in the opposite

direction as if a different mode shape is forming. This

observation is directly related to another interesting

result presented in table 3 and shown in figure 6 for

the plate with diVe" = 0.11; that is, the plate with

d/W = 0.11 buckles at a higher load than tile corresponding plate without a cutout.

The buckle mode shape for the [9010]s plate without a cutout consisted of one half-wave in the direction normal to the loading and two half-waves in
the directionparallelto the loading. The buckle

modeshapefor tile [9010Isplatewith d/W = 0.11

is shown in figure 10 by the moird fringe patterns

photographed during the test. The fringe pattern for

P/P° r = 0.99 indicates that the plate is beginning

to buckle into a mode that lies between the mode

observed for tile plate with d/W = 0 and a mode

consisting of one half-wave in each direction. The

fl'inge pattern for P/P° r = 1.14 shown in figure 10

indicates that as the load increases the plate is moving closer to the mode with two half-waves along the

loading direction. The buckling mode shapes for all

the plates with d/W > 0.11 consisted of one halfwave along both the plate length and width.

To gain insight into this phenomenon, some buckling analyses were performed by using the BUCKO

computer code described in reference 12. Analytical results were obtained for [90m]s plates, based on

nominal lamina properties and ply thicknesses; for

plate aspect ratios ranging from L/W = 0.4 to 2.0;

and for cutout sizes of d/W = 0 and 0.10. These

analytical results are shown in figure 11.

The analytical results shown in figure l l are the

usual festooned curves that relate buckling coefficient

K to plate aspect ratio L/W. At the first set of

cusps in these curves, the buckle mode changes from

one half-wave along the loading direction to two halfwaves. The analytical results indicate that the plate

with L/W = 1 and d/W = 0.10 is on the cusp of the

intersecting dashed-line curve. This analytical result

and the experimental results previously described

suggest that a modal interaction was present during

the test of tile [9010]s plate with d/t'I .... = 0.11.

Moreover, the resulting interaction gave rise to a

slight increase in buckling load.

Results .for [(0/90)5]_ plates. The results presented in table 4 and figure 8 indicate that the prebuckling stiffnesses of the [(0/90)5]s plates decrease

with increasing cutout size, with the exception of the

plate with d/W = 0.42. This plate exhibits a prebuckling stiffness almost equal to that of the plate

with d/W = 0.32. The maximum decrease in prebuckling stiffness is approximately 61 percent for the

plate with d/VV = 0.66.

The results presented in table 4 also indicate that

the buckling loads of the plates decrease monotonically as d/W increases up to 0.32, and then the buckling loads change very littIe as d/VV increases. The

buckling loads of the plate with d/W = 0.32 and 0.66

are approximately 11 percent and 5 percent less than

the buckling load of the corresponding plate without

a cutout, respectively. Analytical results indicating

a somewhat similar trend are also presented in reference 2. The buckling mode shapes for all the plates

consisted of one half-wave along both the plate length

and width.

The results presented in table 4 and figure 8 indicate that the initial postbuckling stiffnesses of the

plates decrease monotonically with increasing cutout

size. The largest decrease is approximately 51 per2

cent for d/W = 0.66. Comparing the prebuckling

stiffness to the postbuckling stiffness given in table 4

for each specimen indicates that as the cutout size increases, the change in axial stiffness due to buckling

decreases from approximately a 53 percent stiffness

reduction for d/W = 0 to a 31 percent stiffness reduction for d/W = 0.32. The stiffness reduction for

d/W = 0.66 is approximately 41 pcrcent.

The curves of nondimensional load versus transversc deflection shown in figure 9 for the [(0/90)5]s

platcs indicate that the plates with d/W = 0.32 and

0.21 had the largest initial imperfections. The results

presented in this figure for the plate with d/W = 0.66

indicate that the plate initially deforms in the direction of its geometric imperfection shape and then

buckles into a similar mode shape in the opposite direction. This behavior is manifested in figure 8 by the

small jump in the curve of load versus end-shortening

(similar to what is encountered when a change in

buckle mode occurs) ncar the buckling load for the

plate with d/W = 0.66.

Symmetrically Laminated Angle-Ply

Plates

Experimental results were also obtained for

[(+30)6]s, [(4-60)6]_, and [(4-45)6]_ square plates having up to five different cutout sizes ranging from

d/W = 0 to 0.66. The cutout size, buckling load,

prebuckling stiffness, and postbuckling stiffness of

each [(±30)aJs, [(±60)6]s, and [(±45)6Js plate are presented in tables 5, 6, and 7, respectively. Curves

of nondimensional load versus end-shortening and

nondimensional load versus transverse deflection are

prescnted in figures 12 and 13, respectively, for the

[(+30)6]s plates; in figures 14 and 15, respectivcly,

for the [(±60)6]s plates; and in figures 16 and 17,

respectively, for the [(±45)6]s plates.

Results for [(+30)6]s plates. The results presented in table 5 and figure 12 indicate that the prebuckling stiffnesses of the [(±30)6]s plates decrease

monotonically with increasing cutout size. The maximum decrease in prebuckling stiffness is approximately 54 percent for the plate with d/W = 0.66.

The results presented in table 5 also indicate that

the buckling loads of the plates decrease monotonically with increasing cutout size up to d/W = 0.60.

The buckling load of the plate with d/W = 0.60 is

6
approximately16percentlessthanthebucklingload

of the correspondingplatewithout a cutout. The

platewith d/W = 0.66 exhibits a buckling load approximately 13 percent less than the buckling load of

the corresponding plate without a cutout. Analytical results that indicate a somewhat similar trend are

presented in reference 3. The buckling mode shapes

for all the plates consisted of one half-wave along

both their length and width.

The results presented in table 5 and figure 12

indicate that the initial postbuckling stiffnesses of

the plates with d/W = 0.11 and 0.32 are approximately 4 percent higher than tile postbuckling stiffness of the plate with d/W = 0. The plates

with d/W = 0.60 and 0.66 exhibit initial postbuckling stiffnesses that are, respectively, approximately

16 percent and 27 percent less than the stiffness of

the corresponding plate without a cutout. Comparing the prebuckling stiffness with the postbuckling stiffness given in table 5 for each plate indicates

that, as the cutout size increases , the change in axial stiffness due to buckling decreases monotonically

from approximately a 66-percent stiffness reduction

for d/W = 0 to a 46-percent stiffness reduction for

d/W = 0.66.

The curves of nondimensional load versus transverse deflection shown in figure 13 indicate that the

plates with d/I¥ = 0.11 and 0 had the largest initial imperfections. The results also suggest, that the

region adjacent to the cutout in the buckled plates

with d/W < 0,32 becomes more flexible in bending

as the cutout size increases. The results presented in

this figure for the plates with d/IV = 0.60 and 0.66

indicate that the plates failed at loads much lower in

the postbuckling range than the other plates.

Results for [(+60)6]s plates. The results presented in table 6 and figure 14 indicate that the

prebuckling stiffnesses of the [(+60)6]s plates decrease monotonically with increasing cutout size.

The maximum decrease in prebuckling stiffness is

approximately 60 to 69 percent for the plate with

d/W = 0.66. The curve of nondimensional load versus end-shortening shown in figure 14 for the plate

with d/W = 0.66 exhibits a nonlinear prebuckling

path. Neglecting the nonlinearity gives a prebuckling

stiffness reduction due to the cutout of approximately

60 percent, whereas including the nonlinearity accounts for another 9-percent reduction in stiffness.

Because of the nonlinear path, the buckling load was

estimated with the corresponding curve of nondimensional load versus transverse deflection presented in

figure 15. At the buckling load, the secondary branch

of the curve of load versus end-shortening is linear.

Strain-gauge data (for back-to-back pairs of gauges

located near the edge of the cutout) recorded during

testing of the plate with d/W = 0.66 corroborated

the presence of material nonlinearity. This fact is illustrated in figure 18 by the nonlinear shape of the

initial part of the curves giving nondimensional load

versus axial strain. The strain-gauge data for the

plates with the smaller cutout sizes showed no indication of material nonlinearity prior to buckling.

The results presented in table 6 also indicate that

the buckling loads of the plates decrease monotonically with increasing cutout size up to d/W = 0.32.

The buckling Ioad of the plate with d/W = 0.32 is approximately 16 percent less than the buckling load of

the corresponding plate without a cutout. The plate

with d/W = 0.66 exhibits a buckling load 23 percent

greater than the buckling load of the corresponding

plate without a cutout. These buckling loads exhibit

a trend that is similar to the analytically obtained

buckling loads presented in reference 3. The buckling mode shapes for all the plates consisted of one

half-wave along both the plate length and width.

The results presented in tables 6 and figure 14

indicate that the initial postbuckling stiffnesses of

the plates decrease with increasing cutout size. The

largest reduction in stiffness is exhibited by the plate

with d/W = 0.66, and is approximately 74 percent

of the stiffness of the corresponding plate without

a cutout. The plates with d/W = 0 and 0.11 also

exhibit a change in buckle pattern from one half-wave

along the loading direction to two half-waves along

the loading direction at approximately P/P_° r = 1.4.

Associated with these changes in buckle pattern are

additional reductions in postbuckling stiffness of 26

and 34 percent of the corresponding prebuckling

stiffnesscs for the plates with d/W = 0 and 0.11,

respectively. Comparing the prebuckling stiffness

with the postbuckling stiffness given in table 6 for

each specimen with d/IV < 0.66 indicates that as

the cutout size increases, the axial stiffness due to

initial buckling decreases about 33 to 39 percent. The

plate with d/W = 0.66 cxhibits a stiffness reduction

(including thc effects of material nonlinearity) of

approximately 60 percent.

Thc curves of nondimensional load versus transvcrse deflection shown in figure 15 suggest that the

region adjacent to the cutout (where the displacement was measured) becomes more flexible in bending as the cutout size increases for all cutout sizes

except d/W = 0.66. The results in this figure also

show a change in buckle pattern for the plates with

d/W = 0 and 0.11 at approximately P/P° r = 1.4.

Results for [(_45)61s plates. The results

presented in table 7 and figure 16 indicate that

the prebuckling stiffnesscs of the [(=k45)6]s plates
decreasemonotonicallywith increasingcutoutsize.

The maximumdecreasein prebucklingstiffnessis

approximately62 to 88percentfor the platewith

d/W = 0.66. The curve of nondimensional load versus end-shortening shown in figure 16 for the plate

with d/W = 0.66 exhibits a substantial nonlinear

prebuekling path. Neglecting the nonlinearity gives

a prebuckling stiffness reduction due to the cutout

of approximately 62 percent, whereas including the

nonlinearity accounts for another 26-percent reduction in stiffness.

Because of the nonlinear prebuckling path of the

plate with d/W = 0.66, the buckling load was estimated with the corresponding curve of nondimensional load versus transverse deflection presented in

figure 17. At the buckling load, the secondary branch

of the curve of load versus end-shortening is not linear like that of the [(+60)6]s plate with d/W = 0.66.

Placing a tangent, line through the buckling load indicates that the initial postbuckling stiffness and the

prebuckling stiffness just prior to buckling are the

same. This observation suggests that significant nonlinear material behavior was present, during buckling.

Strain-gauge data (from back-to-back pairs of gauges

located near the edge of the cutout) recorded during

testing of the [(+45)6]s plates indicate the presence

of very small amounts of material nonlinearity in the

plates with d/W < 0.66. These small amounts of

material nonlinearity account for the shape of the

curves of nondimensional load versus end-shortening

shown in figure 16; that is, the curves start out linear

and then become substantially nonlinear just after

buckling.

The results presented in table 7 also indicate

that the buckling loads of the [(±45)6]_ plates decrc_e monotonically with increasing cutout size up

to d/W = 0.32. The buckling loads of the plate

with d/W = 0.32 and 0.66 are approximately 14 percent less and 14 percent greater, respectively, than

the buckling load of the corresponding plate without a cutout. For all the plates, the buckling load

was estimated from the appropriate curve of load versus transverse deflection and associated strain-gauge

data due to the rapid onset of nonlinear behavior immediately after buckling. The buckling mode shapes

for all the plates consisted of one half-wave along

both the plate length and width.

The results presented in tables 7 and figure 16

indicate that the initial postbuckling stiffnesses of

the plates decrease with increasing cutout size. The

largest reduction in stiffness is exhibited by the plate

with d/W = 0.66 and is approximately 73 percent

(including substantial material nonlinearity) of the

stiffness of the corresponding plate without a cutout.

ComParing the prebuckling stiffness to the postbuckling stiffness given in table 7 for each [(+45)6]s plate

with d/W < 0.66 indicates that as the cutout size

increases, the reduction in axial stiffness duc to initial buckling is around 45 to 55 percent. The plate

with d/W = 0.66 exhibits a stiffness reduction, of approximately 68 percent that includes the effects of

material nonlinearity. An important point to note is

that the postbuckling stiffness presented herein for

each specimen is dependent upon the estimate of the

buckling load due to the nonlinear shape of the curves

of nondimensional load versus end-shortening.

The curves of nondimensional load versus transverse deflection shown in figure 17 for the [(=1=45)6ts

plates with d/W < 0.66 indicate that the region adjacent to the cutout (where the displacement was measured) of the buckled plates becomes more flexible in

bending as the cutout size increases. The results presented in this figure for the plate with d/W = 0.32

indicate that the plate initially deforms in the direction of its geometric imperfection shape and then

buckles into a similar mode shape in the opposite direction. However, no indication of this behavior is

given by the corresponding curve of load versus endshortening. The plate with d/W = 0.66 failed before

getting very far into the postbuekling range.

Failure Tests

Most of the specimens tested in the experimental

study described in this paper were loaded gradually

to approximately twice their buckling loads. However, some of the specimens were loaded to failure.

The particular specimens loaded to failure, their failure loads, and average axial strains at failure are

listed in table 9. The average strains recorded in

table 9 are obtained by dividing the end-shortening

at failure by the true 10-in. length of the plates.

Three [9010]s plates with ratios of cutout diameter

to plate width d/W of 0.32, 0.42, and 0.66 were

loaded to failure. The plate with d/W = 0.32 failed

at the highest load. The plates with d/W = 0.42

and 0.66 failed at loads approximately 19 percent

and 15 percent less, respectively, than the failure

load of the plate with d/W = 0.32. Similarly,

the plates with d/W = 0.42 and 0.66 failed at

average strains approximately 40 percent smaller and

17 times larger, respectively, than the failure strain

of the plate with d/W = 0.32. In all cases, the

[9010]s specimens failed at the net section of the

plate, along the fibers in a matrix failure mode.

The curves of nondimensional load versus transverse

deflection presented in figure 7 for the [9010]_ plates

with d/W = 0.42 and 0.66 show an abrupt change in

bending stiffness near the cutout that is associated

with the onset of matrix cracking. Moreover, the

curves shown in figure 7 indicate that all three of
theplatesfailedat roughlythesamemagnitudesof

transversedeflection.

Two [(+30)6]splates d/W = 0.60 and 0.66

were loaded to failure. The failure loads and average

failure strains for the two plates are nearly the same.

The plate with d/W = 0.60 failed along a line that

is parallel to the +30 ° fibers and runs from the edge

of the cutout at the net section to the unloaded edge

of the plate. The specimen appeared to have a clean

break of the -30 ° fibers along the failure line just

described. The plate with d/W = 0.66 failed along a

line that is parallel to the +30 ° fibers and runs from

the edge of the cutout to the corner of the plate. This

specimen also appeared to have a clean break of the

-30 ° fibers along the failure line just described but

did not intersect the cutout at the net section.

One [(+60)6]s plate d/W = 0_66 was loaded

to failure. The failure load of the [(=k60)6]s plate was

approximately 29 percent less than the failure load

of the [(+30)6]s plate. However, tile average failure

strain of the [(i60)6]s plate was approximately 3.3

times that of the [(:t:30)6]s plate. In the [(+60)6]s

plate, the failure appeared to be due to delaminations

that started at the unloaded edges of the plate near

the net section and propagated to the free edge of

the cutout. The delaminations also appeared to

propagate along the +60 ° fiber direction.

One [(+45)6]s plate d/lV = 0,66 was also

loaded to failure. The failure load of the [(+45)6]s

plate was approximately 4 percent higher than the

failure load of the [(+30)6]s plate, and the average failure strain of the [(:t:45)6]s plate was approximately 3.6 times larger than that of the [(+30)6]s

plate. In the [(±45)6]s plate, the failure also appeared to be due to delaminations that started at

the unloaded edges of tile plate near the net section

and propagated to the free edge of the cutout. The

delaminations also appeared to propagate along the

+45 ° fiber direction.

Discussion of Results

Overall stiffness trends. The experimental results presented in this paper include a wide range

of cutout sizes and a broad spectrum of plate orthotropy (in addition to plate isotropy). Only one

specimen was tested for each combination of cutout

size and plate orthotropy. Due to the limited amount

of testing on each specimen type, the degree of scatter in the experimental data is not well-known. However, the experimental data presented in this paper

are useful in identifying overall trends exhibited by

each family of plates studied. To indicate the overall

trends, results showing the reduction in prebuckling

stiffness due to cutouts, results showing the change

in postbuckling stiffness due to cutouts, and results

showing the change in axial stiffness the plates experience in going from an unbuckled state to a buckled

state are presented in tables 10, 11, and 12, respectively, for all 40 specimens tested.

The results presented in table 10 indicate the reduction in axial stiffness prior to buckling with respect to the prebuckling stiffness of the corresponding plate without a cutout as a function of inplane

plate orthotropy. The results presented in this table

indicate a complex interaction between cutout size

and plate orthotropy on the stiffness reduction. For

example, the [010]s plates lose a substantial amount

of inplane stiffness (approximately 27 percent) when

small cutouts are present, whereas the [90m]s and

[(+30)G],_ plates retain a substantial amount of stiffness (approximately 88 percent) until d/W becomes

greater than 0.32. The largest stiffness reductions are

exhibited by the [010],_ and [(0/90)5]._ plates for tile

flfll range of cutout sizes. The smallest stiffness reductions are exhibited by the [9010Is plates for most

of the cutout sizes. In all cases, the reduction in axial

stiffness due to the cutout is generally not equal to

the reduction in cross-sectional area at the net section of the plate.

The results presented in table 11 indicate the reduction in initial postbuckling stiffness, with respect

to the initial postbuckting stiffness of the corresponding plate without a cutout, as a function of cutout

size and plate orthotropy (inplane and bending orthotropy). These results also indicate a complex interaction between cutout size and plate orthotropy

that influences the change in postbuekling stiffness.

The general trend for the most part is a monotonic

reduction in postbuckling stiffness with increasing

cutout size. The isotropic and the [(0/90)5]s plates

exhibit the largest losses in postbuckling stiffness for

cutout sizes up to d/tV = 0.60. The [(-t-45)6]s and

[(+60)6]s plates with d/W = 0.66, which deform inelastically prior to buckling, exhibit the largest reductions in postbuckling stiffness. In contrast, the

[9010]_ plates with d/W = 0.21, 0.32, and 0.42 exhibit

postbuckling stiffnesses between 25 and 56 percent

higher than the postbuckling stiffness of the corresponding plate without a cutout. The magnitude of

these increases suggests that the increases in stiffness

noted are not due entirely to scatter in the expcrimental data and suggest that further investigation

of the importance of plate orthotropy on the postbuckling behavior of plates with cutouts should be

performed. Moreover, these results suggest the possibility of tailoring the cutout size and the stacking

sequence of a composite plate to optimize postbuckling stiffness.

9
Theresultspresentedin table12indicatethereductionin axial stiffnessassociatedwith changing

fromanunbuckledstateto abuckledstateasafunctionofcutoutsizeandplateorthotropy.Thestiffness

changespresentedinthistablecorrespondtothepercentagedifferencebetweenthe prelmcklingstiffness

and the initial postbucklingstiffnessof eachplate.

Theresultspresentedin this tableindicatethat the

isotropieplateslosebetween35 and 40percentof

their stiffnessdueto bucklingfor the flfll rangeof

cutout,sizes.Thespeciallyorthotropicplates,however,havea muchlargerspreadin the stiffnessreductionswith respectto the cutoutsizestlian the

isotropicplates.Tile [(±60)6]splatesexhibita trend

similarto thatoftheisotropicplates( lessthan6percentvariation)with theexceptionof the platewith

d/W = 0.66. These plates deform inelastically prior

to buckling. The largest stiffness reduction is exhibited by the [010]s plate without a cutout, and the

smallest stiffness reduction is exhibited by the [9010]s

plate with d/I_V = 0.21.

Behavior of out-of-plane displacement

field. Results showing the effects of cutout size

and plate orthotropy on the shape of the out-ofplane displacement field (transverse deflection) arc

presented in figure 19 for [010]s, [(0/90)51s, [90101s,

and [(±60)6]s plates with d/W = 0.11, 0.32, 0.42,

and 0.66. The out-of-plane displacement contours

shown in this figure arc photographs of moir_ fringe

patterns that were taken during testing and correspond to load levels that were well into the postbuckling range. The out-of-plane displacement patterns

shown in the figure consist of onc half-wave in both

the length and width directions of the plate with thc

exception of the [90t0]s plate with d/W = 0.11. In

this case, the displacement pattern consists of two

half-waves along the plate length and one half-wave

across the plate widthl

The results shown in figure 19 indicate that

cutout size and plate orthotropy have a pronounced

effect on the distributions of the out-of-plane displacements of the plates. The results shown in figure 19(a) for the [0m]_ plate with d/W = 0.11 indicate that lines of constant out-of-plane displacement

are somewhat oval in appearance but substantially

elongated, with the longest axis of the oval being

normal to the loading direction. The out-of-plane

displacement distributions are typical of those that

usually occur in a highly orthotropic plate. As the

cutout size inceases, the out-of-plane displacement

The results shown in figure 19(b) for the [(0/90)5]s

plates (plates with a more moderate degree of

orthotropy) indicate out-of-plane displacement patterns that arc more evenly distributed than the displacement patterns of the [0to]s plates. The displacement patterns shown for the [(0/90)5]s plates with

d/W = 0.i1 and 0.42 are typical of an overall plate

type of bending mode as opposed to a mode in which

the out-of-plane displacement field becomes concentrated in the lateral regions of the plate near the unloaded edges. For the cutout size of d/W = 0.66, the

out-of-plane displacement field in the [(0/90)5]s plate

also becomes concentrated in the lateral regions of

the plate near the unloaded edges but not to the extent that is exhibited by the highly orthotropic [010]s

plates. Results obtained for thc isotropic plates and

[(+30)6]s plates indicated behavior very similar to

the behavior of the [(0/90)5]s plates.

The results presented in figure 19(c) for the [9010]s

plates indicate out-of-plane displacement patterns

that are evenly distributed. The results shown in

figure 19(c) for the [9010]_ plate with d/W = 0.66

appear to be more evenly distributed over tile plate

than the corresponding results for the [010]s and

[(0/90)5]s plates, but in addition, the lines Of Constant out-of-plane displacement along the edges of

the cutout appear to be more densely spaced.

The results presented in figure 19(d) for the

[(+60)6]._ plates indicate out-of-plane displacement

patterns that are somewhat, evenly distributed for

the plates with d/W = 0.11 and 0.32. However, the

results shown in figure 19(d) for the [(+60)6]s plate

with d/IV = 0.66 indicate that tile out-of-plane displacement field is much more localized around the

vicinity of the net section of the plate, unlike the

out-of-plane displacement fields of the other plates

with the large cutouts. This more localized distribution of the out-of-plane displacements may be related to the fact that the [(±60)6]s stacking sequence

tends to relocate the inplane load outboard of the

cutout much more than the isotropie, specially orthotropic, and [(±30)6]_ plates relocate the inplane

load. This more localized distribution of the out-ofplane displacements may also be related to the nonlinear (material) prebuckling deformations exhibited

by this plate. The corresponding [(±45)6]_ plates exhibited behavior very similar to the behavior of the

[(±60)6]s plates.

Modal interaction. Another important topic

briefly investigated in this paper that may be important in prelinfinary design of structures is the relafield redistributes to become more concentrated in tionship between cutout size and modal interaction in

the lateral regions of the plate near the unloaded plates. The results presented in figures 6, 7, 10, and

edges, and fully envelops the full length of the plate. 11 for the [9010]s plates indicate that increasing the

10

E
cutout size can cause a buckling mode shape change

to occur in a plate. For example, the [9010]s plate

without a cutout buckled into a mode that consisted

of one half-wave normal to the loading direction and

two half-waves in the loading direction, whereas the

corresponding plate with d/W = 0.21 buckled into

a mode that consisted of one half-wave in each direction. Moreover, the results show that a cutout

size can exist for which a modal interaction occurs at

the onset of buckling. Specifically, the [9010Is plate

with d/W = 0.11 exhibited an interaction between

the mode shapes of the corresponding plates with

d/W = 0 and d/W = 0.21. (See fig. 10.) None of the

other 39 plates tested exhibited modal interaction at

the onset of buckling.

Insight into the behavior of the plates with

cutouts tested can be obtained by examining the

buckling behavior of rectangular orthotropie plates

that are simply supported on the unloaded edges, in

accordance with the test fixture used in the experimental study. The standard plot of buckling coefficient K versus plate aspect ratio L/W found in most

text books on stability of plates is composed of a series of curves, referred to as festooned curves, that

intersect at cusps. Two examples are shown in figure 11. Each independent curve between cusps represents a specific buckling mode for a certain group

of plate aspect ratios. As the plate aspect ratio increases past a certain value, tile number of longitudinal half-waves in the buckling mode changes. Furthermore, at the cusps a unique buckling mode is

indeterminant, since two distinct modes possess the

same buckling load.

The location of thesc cusps is directly related to

buckling behavior of the corresponding infinitely long

plate. Analysis shows that the minimum points on

the festooned curves occur at integer multiples of the

aspect ratio of the repetitive buckle mode of the infinitely long plate. For other aspect ratios, the plate

buckles at a higher load. This relationship accounts

for the festooned nature of the curves of buckling

load versus plate aspect ratio. The particular value

of the aspect ratio of the buckling mode shape of

the infinitely long plate is directly related to the inplane stresses acting in the plate, tile boundary conditions acting on the unloaded edges, and the plate

bending orthotropy. For the simply supported [9010]s

and [010]s plates without cutouts investigated in the

present study, the aspect ratios of tile buckling mode

of infinitely long [9010]s and [010]s plates are approximately 0.54 and 1.85, respectively. Thus, minimum

points and cusps in a festooned curve plot of buckling

coefficient versus plate aspect ratio are more closely

spaced for [9010]s plates than for [010Is plates. The

close spacing of the cusps of the festooned curves of

the [9010]s plates and the close proximity of the first

cusp to tile unit plate aspect ratio (see fig. 11 for

d/W = 0) indicate that a [9010]s square plate is more

likely to change buckle mode shape when a small perturbation in the buckle mode aspect ratio occurs than

the other square plates investigated herein.

In a [9010Is plate with d/W = 0.11, the cutout

causes redistribution of tile inplane prebuckling

stresses and causes a decrease in bending stiffness

in the central region of the plate. The decrease in

bending stiffness results from the fact that material

has been removed from the plate and from the fact

that a free interior boundary is present. These effects

(inplane and bending) associated with the cutout result in perturbing the aspect ratio of the buckle (or

buckles) forming the mode shape and effectively shift

the festooned curves until the first cusp is at a plate

aspect ratio of 1, as indicated by the dashed line

in figure 11. As the cutout size increases, the associated festooned curves shift more until the first

cusp occurs at a plate aspect ratio significantly larger

than 1. This observation accounts for the fact that

the [9010]s plates with d/W > 0.11 had buckle modes

that consisted of one half-wave in each direction.

Analysis indicates that the other plates investigated in this study all possess festooned curves in

which the first cusp, corresponding to a change in

buckle pattern from one longitudinal half-wave to

two half-waves, occurs at a plate aspect ratio substantially larger than 1. Specifically, the first cusp

for these plates occurs at aspect ratios far enough

removed from a value of 1 such that perturbations

in the aspect ratio of the buckle mode due to cutout

sizes as big as d/W = 0.66 do not cause a change

in mode shape or a modal interaction at the onset

of buckling. This observation accounts for the fact

that the other 39 plates tested buckled into distinct

mode shapes that consisted of one half-wave in each

direction for the full range of cutout sizes. Analytical results supporting this discussion can be found

in references 1 and 3 for specially orthotropic plates

and symmetrically laminated angle-ply plates with

cutouts, respectively. The results reported in these

references, and results used to support the previous

discussion, were obtained with the computer program

described in reference 12.

Anomalous behavior. Results were obtained

for the [0m]s plate with d/W = 0.32 that indicated buckling behavior atypical of orthotropic

plates. More specifically, the results presented in figures 4 and 5 for this plate indicate that the plate

deforms initially in the direction of its geometric

imperfection shape and then rapidly buckles into a

similar mode shape in the opposite direction. During the testing of this plate, a snapping sound was

11
heardwhenthe platebuckled.Thisobservationis

supportedby thejump in the correspondingcurve

of loadversusend-shorteningshownin figure4 and

by tile discontinuityin thecorrespondingcurveload

versustransversedeflectionshownin figure5. These

resultsindicatethat.theonsetof bucklingoccurred

in amannersimilarto a changein bucklepattern.

For this particulartest specimen,it wasfound

that theloadededgesof tile specimenhadnot been

properlymachinedwhichresultedin an eccentric

loading. Tile eccentricity,producedbendingmomentsthat increasedin magnitudeasthe applied

loadingincreasedand actedto movethe plate in

the directionoppositeto the geometricimperfection. Bucklingoccurredwhenthesebendingmomentsovercamethetendencyof theplateto deform

in thedirectionof its geometricimperfection.

Similarbehaviorwasexhibitedby the [(0/90)5]s

plate with d/W = 0.66. The curve of load versus

transverse deflection shown in figure 9 for this case

did not have a sharp discontinuity like that shown

in figure 5 for the [0m]s plate with d/W = 0.32.

However, a jump is present in the curve of load versus

end-shortening shown in figure 8 for the [(0/90)5]s

plate with d/W = 0.66, and a snapping sound was

heard during the test when the plate buckled.

The curve of load versus transverse deflection

shown in figure 17 for [(±45)6]s plate with d/W =

0.32 indicates that the plate also deformed initially in

the direction of its geometric imperfection. However,

no jump in the corresponding curve of load versus

end-shortening (see fig. 16) is present, and no snapping sound was heard when the plate buckled. The

absence of these effects may" be associated with the

nonlinear material behavior.

In the investigation of buckling and postbuckling

behavior of asymmetrically laminated plates, such

as the work reported in reference 13, th e !nherent

mechanical coupling between membrane and bending actio n in the plate produces out-of-plane deformations as the applied axial loading increases in a

manner similar to that associated with tile loadpath eec_entricit_52pre_;iously described. Tile - !nte nsity of the prebuckling moments due to mechanical

coupling, and the direction of the associated outof-plane deformation, are determined by the plate

staek{ng sequence. Whc_n-the prebuckilng mOments

produce out-of-plane deformations that act in a direction opposite to the plate's geometric imperfection, a snap-through buckling similar to that exhibited by tile [010]s plate with d/W = 0.32 may occur.

In such cases it is important to understand the two

mechanisms, mechanical coupling (material induced

eccentricity) and edge effects (geometry induced eccentricity), responsible for the plate behavior.

12

Concluding Remarks

An experimefital study of the postbuckling behavior of square compression-loaded graphite-epOxy

plates and isotropic plates with central circular

cutouts has been presented. A wide range of cutout

sizes and a broad spectrum of plate orthotropy has

been investigated. Specifically, results have been

presented for unidirectional [0m]s and [90m]._ plates,

[(0/90)5]s plates, and for aluminum plates. Results

have also been presented for [(+0)6]s angle-ply plates

for values of 0 = 30 °, 45 °, and 60 °. The ratios of circular cutout diameter to plate width ranged from 0

to 0.66.

Experimental results have been presented that indicate that tile change in axial stiffness of a plate at

buckling is strongly dependent upon cutout size and

plate orthotropy. The presence of a cutout gives rise

to an internal load distribution that changes, sometimes dramatically, as a function of cutout size coupled with the plate orthotropy. Moreover, results

that have been presented indicate that tile reduction in prebuckling axial stiffness, associated with

a cutout, is not generally equal to the reduction in

cross-sectional area of the plate at the net section.

In the buckled state, the role of orthotropy increases to include bending in addition to membrane

orthotropy. Experimental results indicate that most

of the plates tested with cutouts exhibit, less postbuckling stiffness than the corresponding plate without a cutout and that the amount of postbuckling

stiffness generally decreases with increasing cutout

size. However, the experimental results also indicate that some of the highly orthotropic plates with

cutouts exhibit higher postbuekling stiffness than the

corresponding plate without a cutout. In all these

cases, the cutout size and plate orthotropy was found

to dramatically affect the distribution of the out--

of-plane displacement field that occurs in a buckled

plate. These results suggest that the complex interaction of cutout size and orthotropy on the internal

load distribution in plates needs further investigation. These results also suggest that the cutout size

and the stacking sequence of a composite plate could

be tailored to optimize postbuckling stiffness.

Experimental results also indicate that a cutout

can influencemodai interaction in a plate. Specifically, results show a plat(; with a relatively small

cutout buckling at a higher load than the corresponding plate without a cutout due to modal interaction.

Other results indicate the presence of nonlinear prebuckling deformations, due to material nonlinearity,

in the angle-ply plates with 0 =_45 ° and 60 °. The

nonlinear prebuckling deformations are more pronounced in the plates with 0 = 45 °, and become
even more pronounced as the cutout size increases.

An important finding of this experimental study is

that plates with large radius cutouts do exhibit some

postbuckling strength.

NASA Langley Research Center

Hampton, VA 23665-5225

June 29, 1990

References

1. Nemeth, Michael Paul: Buckling Behavior of Orthotropic

Composite Plates With Centrally Located Cutouts. Ph.D.

Diss., Virginia Polytechnic Inst. & State Univ., May 1983.

2. Nemeth, Michael P.; Stein, Manuel; and Johnson, Eric R.:

An Approximate Buckling Analysis for Rectangular Orthotropic Plates With Centrally Located Cutouts. NASA

TP-2528, 1986.

3. Nemeth, Michael P.: Buckling Behavior of CompressionLoaded Symmetrically Laminated Angle-Ply Plates With

Holes. AIAA J., vol. 26, no. 3, Mar. 1988, pp. 330 336.

4. Yu, \Vei-_Ven; and Davis, Charles S.: Buckling Behavior and Post-Buckling Strength of Perforated Stiffened

Compression Elements. The First Specialty Conference on

Cold Formed Steel Structures, Wei-Wen Yu, ed., Civil Engineering Dept., Univ. of Missouri-Rolla 1971, pp. 58 64.

5. Martin, James: Buckling and Postbuckling of Laminated

Composite Square Plates With Reinforced Central Circular Holes. Ph.D. Diss., Case \Vestern Reserve Univ., 1972.

6. Yu, Wei-Wen; and Davis, Charles S.: Cold-Formed Steel

Members With Perforated Elements. J. Struct. Div.,

American Soc. Cir. Eng., vol. 99, no. ST10, Oct. 1973,

pp. 2061 2077.

7. Ritchie, D.; and Rhodes, J.: Buckling and Post-Buckling

Behavior of Plates With Holes. Aeronaut. Q., vol. 26,

pt. 4, Nov. 1975, pp. 281 296.

8. Starnes, James H., Jr.; and Rouse, Marshall: Postbuckling and Failure Characteristics of Selected Flat

Rectangular Graphite-Epoxy Plates Loaded in Compression. A Collection of Technical Papers AIAA/ASME/

ASCE/AHS 22nd Structures, Str_tcturat Dynamics _4 Materials Conference, Part 1, Apr. 1981, pp. 423 434. (Available as AIAA-81-0543.)

9. Roberts, T. M.; and Azizian Z. G.: Strength of Perforated Plates Subjected to In-Plane Loading. Thin-Walled

Structures, vol. 2, no. 2, 1984, pp. 153 164.

10. Narayanan, R.; and Chow, R. Y.: Ultimate Capacity of

Uniaxially Compressed Perforated Plates. Thin-Walled

Structures, vol. 2, no. 3, 1984, pp. 241 264.

11. VandenBrink, Dcnifis J.; and Kamat, Manohar P.: PostBuckling Response of Isotropic and Laminated Composite

Square Plates With Circular Holes. Fifth International

Conference on Composite Materials, W. C. Harrigan, Jr.,

J. Strife, and A. K. Dhingra, eds., Metallurgical Soc., Inc.,

c.1985, pp. 1393 1409.

12. Nemeth, Michael P.: A Buckling Analysis for Rectangular Orthotropic Plates With Centrally Loeatcd Cutouts.

NASA TM-86263, 1984.

13. Jensen, David W.; and Lagace, Paul A.: Influence of

Mechanical Couplings on the Buckling and Postbuckling

of Anisotropic Plates. AL4A J., vol. 26, no. 10, Oct. 1988,

pp. 1269 1277.

. 13
Table 1. Experimental Buckling Loads, Prebuckling Stiffnesses, and Initial Postbuckling Stiffnesses for Isotropic Plates

[All plates buckled into one half-wave along their length and width]

Specimen

A1

A2

A3

A4

A5

A6

A7

d/W _

0

.10

.20

.30

.40

.50

.60

Cutout

diameter, in.

0

.95

1.90

2.85

3.80

4.75

5.70

Buckling

load, lb

1872

1828

1736

1656

1808

2007

1995

Prebuckling

stiffness, lb/in.

541 139

477 341

464 290

426445

390 573

326186

312527

Initial

postbuckling

stiffness, lb/in.

329 201

309093

301 479

277 715

244 838

204 760

188 885

aWidth equals distance between test fixture supports, 9.5 in.

Table 2. Experimental Buckling Loads, Prebuckling Stiffnesses, and Initial Postbuckling Stiffnesses for [010Is Laminates

[All plates buckled into one half-wave along their length and width]

Specimen

B1

B2

B3

B4

B5

B6

B7

dlB ....

0

.11

.21

.32

:42

.60

.66

Cutout

diameter, in.

0

1.00

2.00

3.00

4.00

5.70

6.25

Buckling

load, Ib

9256

8975

8767

7689

6842

6464

6158

Prebuckling

stiffness, lb/in.

2 020 460

1473 710

1 194 500

I 266500

724460

641878

526781

Initial

postbuckling

stiffness, lb/in.

481 792

487362

458 587

494 835

394 351

376 653

365660

aWidth equals distance between test fixture supports, 9.5 in.

14
Table3.Experimental BucklingLoads, Prelmckling Stiffnesses, andInitialPostbuekling Stiffnesses for[9010]s Laminates

[Allplatesbuckledintoonehalf-wave alongtheirlengthandwidthunlessotherwise noted]

Specimen

C1

C2

C3

C4

C5

C6

d/W a

0

.11

.21

.32

.42

.66

Cutout

diameter, in.

Buckling

load, lb

0 b2292

1.00 e294

2.00 2041

3.00 1690

4.00 1607

6.25 1868

Prebuckling

stiffness, lb/in.

133066

132 428

120639

125 131

154 529

60955

Initial

postbuckling

stiffness, lb/in.

63445

61114

86 517

79 401

98959

36444

aWidth equals distance between test fxture supports, 9.5 in.

bMode shape consists of two half-waves in the axial direction and one half-wave in the other direction.

CMode shape influenced by modal interaction (see fig. 10).

Table 4. Experimental Buckling Loads, Prebuckling Stiffnesses, and Initial Posttmckling Stiffnesses for [0/905]s Laminates

[All plates buckled into one half-wave along their length and width]

Specimen

D1

D2

D3

D4

D5

D6

D7

dlW _

0

.11

.21

.32

.42

.60

.66

Cutout

diameter, in.

O

1.00

2.00

3.00

4.00

5.70

6.25

Buckling

load, Ib

6950

6729

6407

6207

6510

6467

6581

Prebuckling

stiffness, Ib/in.

955427

646 670

586 874

532 187

535 595

391 619

370703

hfitial

postbuckling

stiffness, lb/in.

447631

397654

391 869

368 960

330118

244 138

220011

aWidth equals distance between test fixture supports, 9.5 in.

15
16

Table 5. Experimental Buckling Loads, Prebuckling Stiffnesses, and Initial Postbuckling Stiffnesses for [( =I=30)6]s Laminates

[All plates buckled into one half-wave along their length and width]

Specimen

E1

E2

E3

E4

E5

d/W a

0

.ll

.32

.60

.66

Cutout

diameter, in.

0

1.00

3.00 "

5.75

6.25

Buckling

load, Ib

10 105

9 850

8 894

8 498

8 745

Prebuekling

stiffness, lb/in.

764 796

755 079

675 404

418 332

353 212

Initial

postbuckling

stiffness, lb/in.

260652

271 228

268 027

217665

191 569

a_Vi(tth equals distance between test fixture supports, 9.5 in.

Table 6. Experimental Buckling Loads, Prebuckling Stiffnesses, and Initial Postbuckling Stiffnesses for [(:t=60)6]s Laminates

JAil plates buckled into one half-wave along their length and width unless otherwise noted]

Specimen

GI

G2

G3

G4

d/W a

0

.ll

.32

.66

Cutout

diameter, in.

0

1.00

3.00

6.25

Buckling

load, lb

579O

5594

4874

7107

Prebuckling

stiffness, lb/in.

202 888

181611

149 485

c81 223 (62 189)

Initial

posthuckling

stiffness, lb/in.

b122920 (71056)

bl21 305 (59 766)

95 804

32199

aV¢idth equals distance between test fixture supports, 9.5 in.

bpostbuckiing Stiffness after change in buckle pattern to two half-waves along the loading axis.

CTangent stiffness at the bifurcation point.
Table7.Experimental BucklingLoads, Prebuckling Stiffnesses, andhfitiaIPostbuckling Stiffnesses for[(-t-45)6]s Laminates

[Allplatesbuckledintoonehalf-wave alongtheirlengthandwidth]

Specimen

FI

F2

F3

F4

d/W _

0

.11

.32

.66

CIltotlt

diameter, in.

0

1.00

3.00

6.25

Buckling

load b, lb

9651

9188

8314

I 1020

hfitial

Prebuckling postbuckling

stiffness, Ib/in. stiffness, lb/in.

388 964 174 152

317 706 174 016

275 011 145 899

c145 951 (46155) 46 155

aWidth equals distance between test fixture supports, 9.5 in.

bBuckling loads estimated from out-of-plane displacements.

CTangent stiffness at the 1)ifllrcation point.

Table 8. Analytic Buckling Loads, Critical End-Shortenings, and Nominal

Thicknesses for Plates Without Cutouts

Plate

type

Aluminum

[Olo1_

[9010Is

[(o/9o)51s

[(+3o)ds

[(±45)6],_

[(_:6o)6]_

Buckling

load,

po, lb

1 773

9 272

2 473

6 544

9 898

10 962

5 944

Critical

end-shortening

O

Act, in.

0.002740

.004556

.014050

.005875

.011759

.029268

.027752

Nominal

thickness

H, in.

0.0647

.1100

.1100

.1100

.1176

.1300

.1176

17
Table 9. Failure Loads and Strains for [9010]s and Angle-Ply Plates

--4

F'---ailure _ Average

d/W a load, lb_ strain, AlL

0.32 t 4028 0.004700

42 / 324 t .oo28oo

.606 / 3412 .080382

[9o1& [(+3o)6]_

Failure Average

load, lb strain, AlL

12 895 0.00421

12 080 .00424

aWidth equals distance between test fixture supports, 9.5 in.

[(±60)6]s

Failure [ Average

__ jo_d, lb [2Fain, A/L

8537 [ 0.014148

[(±45)6]s

Failure Average

load, lb strain, AlL

12601 0.015060

Table 10. Reduction in Axial Prebuekling Stiffness Due to Cutout

Isotropic

d/W a plates

0 0

.10 12

.11

.20 14

.21

.30 21

,32

.40 28

.42

.50 40

.60 42

.66

Reduction in axial prebuckling stiffness, percent

Symmetrically laminated angle-ply plates

0

27

41

37

64

68

74

Specially orthotropic plates

[9010]._

0

1

9

6

b-16

54

[(0/90)5]s [(+30)6]s

0 0

32 1

39

44 t2

44

59 45

61 54

[(+45)608 [(+6o)6]_

0 0

18 10

29 26

c60 (69) 62

°Width equals distance test fixture supports, 9.5 in.

bMinus sign represents increase in stiffness.

CBased on tangent stiffness at the bifurcation point.

18
Table 11. Change in Intial Postbuckling Stiffness Due to Cutout

d/W b

0

.i0

.ii

.20

.21

.30

.32

.40

.42

.50

.60

.66

Change in postbuekling stiffness a, percent

Specially orthotropic plates Symmetrically laminated angle-ply plates

Isotropic

plates

0

-6

-8

-16

-26

-38

-43

[Olo]_ [90lOis [(0/90)5]s

0

-4

+36

+25

+56

0

-11

-12

-18

-26

-45

-51

[(+30)6]_ [(+a5)6]_

0 0

+4 --1

+3 -16

-73

-16

-27

0

+1

-5

+3

-18

-22

-24 -43

[(+60)6]_

0

-1

-22

-74

aChange in stiffness is with respect to the corresponding plate with d/W = 0. Negative values indicate reductions.

bwidth equals distance between test fixture supports, 9.5 in.

Table 12, Reduction in Axial Stiffness in Going From Prebuckling State to hfitial Buckled State

d/W a

0

.10

.11

.20

.21

.30

.32

A0

.42

.50

.60

.66

Reduction in axial stiffness due to buckling, percent

Specially orthotropic plates Symmetrically laminated angle-ply plates

Isotropic

plates

39 76

35

67

35

62

35

61

37

46

37

40 41

31

[010Is [9010]s

52

54

28

37

36

40

[(0/90)5]s

53

39

33

31

38

38

41

[(+30)6]s [(±45)6]s

66 55

64 45

60 47

48

46 b68

[(+60)G]s

39

33

36

b60

aWidth equals distance between test fixture supports, 9.5 in.

bIncludes material nonlinearity.

19
Figure 1. Specimen mounted in test fixture.

Nondimensional

loading, 1.5

P

Per

1.0

I P

I I

1.0 2.0 3.0 4.0 5.0 6.0

Nondimensional end-shortening, _/A_

=

=

m

Figure 2. Nondimensional load versus end-shortening expcrimental results for isotropic square plates with

central circular cutouts.

2O
Location of

transverse

deflection

measurement

II P

3.0

2.0

Nondimensional

loading,

1.0

Location of

transverse

deflection

measurement

for plate with d/W = 0.6

d/W

p 1/ .5 .4 .3 .2 0 .1

.6

I I 1

1.0 2.0 3.0

Nondimensional transverse deflection, 5/H

Figure 3. Nondimensional load versus transverse deflection experimental results for isotropic square plates with

central circular cutouts.

20r

P

N ondimensional / ,,pjj_ _.42.60

_ I I { I I

0 1.0 2.0 3.0 4.0 5.0

o

Nondimensional end-shortening, _?'cr

Figure 4. Nondimensional load versus end-shortening experimental results for [0 ]0]s square plates with central

circular cutouts.

21
2.0 --

1.5

Nondimensional

loading,

P 1.0 L

Per

,5 --

0

-1.0

l P

/-e

71 I-y I/-- Location of

/I v v transverse

_r__ deflection

measurement

w--H d/W

, 'P

0 1.0 2.0

Nondimensional transverse deflection, _/H

I

3.0

Figure 5. NondimensionaI load versus transverse deflection experimental results for [010]s square plates with

central circular cutouts.

. 1.5

Nondimensional

loading, 1.0

_E_P

P_r

.5

1 P

, _ I

2.0 - LI _.._u_ I

d/W

r-0 .11 32

- /p .42Y" .66

- o-7_ _ _

// _ • Buckling

Z_ , Failure-

¢'" I I I I I

0 1.0 2.0 3.0 4.0 5.0

o

Nondimensional end-shortening, A/Acr

I

6.0

Figure 6. Nondimensi0nal load versus end-shortening experimental results for [9010]s square plates with central

circular cutouts.

22
I P

/-e

Nondimensional

loading,

P

"-"6-

Per

Location of

transverse

displacement

d/W

,, -Ip

1-5 t_10 Local /66

I '_k damage /.21 J .,__,i_,l .42

.5 !r I Failure

/

! I I I

0 1.0 2.0 3.0

Nondimensional transverse deflection, 5/H

Figure 7. Nondimensional load versus transverse deflection experimental results for [9010]s square plates with

central circular cutouts.

Nondimensional

loading,

P

P_r

2.0[ d0/_N11 .21 .32 .60 .66

1.5 • .42

1.0

.5

u_ I I 1 I I I

0 1.0 2.0 3.0 4.0 5.0 6.0

Nondimensionafend:shortening, z_,_

Figure 8. Nondimensional load versus end-shortening experimental results for [(0/90)5]s square plates with

central circular cutouts.

23
2.0

1.5

Nondimensional

loading,

p 1.0

po

¢r

.5

0

-1.0

II.P

/-e

_1 ry I/- Location of

/I _' J/transverse

• I _/I deflection

L I _him: I measurement -Lw_LJoo, w

|p

.66_ 6

! ' I i

0 1.0 2.0

Nondimensional transverse deflection, 6/H

J

3.0

Figure 9. Nondimensionat load versus transverse deflection experimental results for [(0/90)5]s square plates

with central circular cutouts.

W

liP F Clamped edge

lip

Simply supported

edge

P

= 1.14

P_r

=,

Strain

gauge

L-90-38

Figure 10. Interacting mode shapes for [9010]s square plates with d/W = 0.11.

i

=

=

=

24
Buckling

coefficient,

K

_

_

' "-;-° .:

--0

--- 0.10

I I I

.5 1.0 1.5 2.0

Plate aspect ratio, L/W

Figure 11. Buckling coefficients for [9010]s rectangular plates with d/W = 0 and 0.10 (loaded edges clamped

and unloaded edges simply supported).

2.0 --

1.5

Nondimensional

loading,

__e 1.o

P_r

.5

11.P

/-e

_ | _ 32

P J_Jl .60

.66

- .11

y I Failure

P" I I 1 I

1.0 2.0 3.0 4.0

Nondimensional end-shortening, &/A°cr

Figure 12. Nondimensional load versus end-shortening experimental results for [( =t=30)6]s square plates with

central circular cutouts.

25
"El W I/-Location of

/I v j/transverse

.'1 _ _/1 deflection

2.o,- "I L)" I measurement

/ 11_1 "_

/ -'-l_-vy-_t r.6o o_oj_.32

Nondimensional I ,,_ JI_/ '°am

P_r 5_

V "--'i ............... ! Failure

I L = l

0 i .O 2.0 3.0

Nondimensional transverse deflection, 8/H

Figure 13. Nondimensional load versus transverse deflection experimental results for [(+30)6]s square plates

with central circular cutouts.

2.0F , _.32

Nondimensional I II / ,'...I

loading, loL =//'// ,,s-, _

 l/f ........

,sI //// • Buckling

I/Z ,F_,,_,o

_" t I I I I I

0 1.0 2.0 3.0 4.0 5.0 6.0

Nondimensional end-shortening, NtL=cr

Figure 14. Nondimensional load versus end-shortening experimental results for [(+60)6]s square plates with

central circular cutouts.

26
2.0

1.5 f

Nondimensional

loading, /

P 1.or

"ii

-1.0

_ /-e ,._ Location of

_1 17 Vtransverse

/I ,,.:.,, /1 deflection

LI ()'/I measurement

0 .11 -"l d d3/_/2

I Failure

I I I

0 1.0 2.0 3.0

Nondimensional transverse deflection, 5/H

Figure 15. Nondimensional load versus transverse deflection experimental results for [(+60)6]s square plates

with central circular cutouts.

P

2.0 r d/W

I_--W_l o

1-5 t Qp _,__.32

Nondimensional l __ // _ 66

loading, loL 7fJ .-"" -.-_""'_'_'"

PP-_cr " L Y_

.s/ /// J • Buckling

I/Z ,F,,,uro

1 ,.I I I I

0 1.0 2.0 3.0 4.0 5.0

Nondimensional end-shortening, A/Acr

Figure 16. Nondimensional load versus end-shortening experimental results for [(=t=45)6]s square plates with

central circular cutouts.

27
li P

-$-_/-- Location of

.11 ,,-L,X tdea_Sv_i r::

:o 21 measurement

1.5

Nondimensional

loading,

p 1.0

.5

_---w--_ d_

- l/- 0.1132

I Failure

1 I I

0 i .0 2.0 3.0

Nondimensional transverse deflection, _/H

Figure 17. Nondimensional load versus transverse deflection experimental results for [(±45)6]s square plates

with central circular cutouts.

2,0 n

1.5

Nondimensional

loading, 1.0

P

P_r

.5

0 I

-.040 .010

II P

Fe

IL__ _ Location of

back-to-back

strain gauges

A&B

A m

,ll

A |P B

B--"

I I I 51

-.030 -.020 -.010 0

Axial strain

Figure 18. Strain near the cutout in the [(+60)6]s square plate with dlW = 0.66.

28
0

II

fifO0

II

II

ob_

13.._

do

IIoo

0 --"

II

oO0

II

2g
3O

II

nn

¢M

T"

II

oO

Q_

II

L

oO


Report Documentation Page

Na_bonal Aer_na_l,qs a_

Space Adm_n_stratlon

I. Report No. t2" Government Accession No. 3. Recipient's Catalog No.

NASA TP-3007 i

4. Title and Subtitle

Buckling and Postbuckling Behavior of Square CompressionLoaded Graphite-Epoxy Plates With Circular Cutouts

7. Author(s)

Michael P. Nemeth

9. Performing Organization Name and Address

NASA Langley Research Center

Hampton, VA 23665-5225 11.

12. Sponsoring Agency Name and Address 13,

National Aeronautics and Space Administration

Washington, DC 20546-0001 14.

5. Report Date

August 1990

6. Performing Organization Code

8. Performing Organization Report No,

L-16777

10, Work Unit No.

505-63-01-08

Contract or Grant No,

Type of Report and Period Covered

Technical Paper

Sponsoring Agency Code

15. Supplementary Notes

Presented at the Eighth DoD/NASA/FAA Conference on

Design, Norfolk, Virginia, November 28 30, 1989.

Fibrous Composites in Structural

16. Abstract

An experimental study of the postbuckling behavior of square compression-loaded graphite-epoxy

plates and isotropic plates with a central circular cutout is presented. Results ark presented for

unidirectional [0m]s and [9010]s plates, [0/905]s plates, and for aluminum plates. Results are also

presented for [(+0)6]s angle-ply plates for values of 0 = 30 °, 45°7 and 60 °. The cxperimcntal results

indicate that the change in axial stiffness of a plate at buckling is strongly dependent upon cutout

size and plate orthotropy. The presence of a cutout gives rise to an internal load distribution that

changes, sometimes dramatically, as a filnction of cutout size coupled with the plate orthotropy.

In the buckled state, the role of orthotropy becomes more significant since bending in addition

to membrane orthotropy is present. Most of the plates with cutouts exhibited less postbuckling

stiffness than thc corresponding plate without a cutout, and the postbuekling stiffness decreased

with increasing cutout size. However, some of the highly orthotropie plates with cutouts exhibited

more postbuckling stiffness than the corresponding plate without a cutout.

17. Key Words (Suggested by Authors(s))

Buckling

Postbuckling

Cutouts

Compression loaded

Composite plates

19. Security Cla.ssif. (of this report)

18. Distribution Statement

Unclassified Unlimited

Subject Category 24

Unclassified 21. No. 31 of Pages122. Price A03

20. Security Cla.ssif. (of this page)

Unclassified

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